Stability of SDOF Nonlinear Viscoelastic System Under the Excitation of Wide-Band Noise

2007 ◽  
Author(s):  
Qinghua Huang ◽  
Wei-Chau Xie
Keyword(s):  
Averaging Method ◽  
Compressive Load ◽  
Wide Band ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Stochastic Bifurcation ◽  
Viscoelastic System ◽  
Band Noise ◽  
Nonlinear Viscoelastic ◽  
Wide Band Noise

The stochastic stability of a single degree-of-freedom (SDOF) nonlinear viscoelastic system under the excitation of wide-band noise is studied in this paper. An example of such a system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The equation of motion is an integro-differential equation with parametric excitation. The stochastic averaging method and averaging method for integro-differential equations are applied to reduce the system. The largest Lyapunov exponents and stochastic bifurcation are studied after the averaged sytem is obtained.

Stochastic Averaging of Strongly Nonlinear Oscillators Under Combined Harmonic and Wide-Band Noise Excitations

2008 ◽  
Vol 130 (5) ◽  
Author(s):  
Y. J. Wu ◽  
W. Q. Zhu
Keyword(s):  
Probability Density ◽  
Wide Band ◽  
Nonlinear Oscillators ◽  
Stochastic Averaging ◽  
Random Excitation ◽  
Stationary Response ◽  
Strongly Nonlinear ◽  
Band Noise ◽  
Wide Band Noise ◽  
Strongly Nonlinear Oscillators

Physical and engineering systems are often subjected to combined harmonic and random excitations. The random excitation is often modeled as Gaussian white noise for mathematical tractability. However, in practice, the random excitation is nonwhite. This paper investigates the stationary response probability density of strongly nonlinear oscillators under combined harmonic and wide-band noise excitations. By using generalized harmonic functions, a new stochastic averaging procedure for estimating stationary response probability density of strongly nonlinear oscillators under combined harmonic and wide-band noise excitations is developed. The damping can be linear and (or) nonlinear and the excitations can be external and (or) parametric. After stochastic averaging, the system state is represented by two-dimensional time-homogeneous diffusive Markov processes. The method of reduced Fokker–Planck–Kolmogorov equation is used to investigate the stationary response of the vibration system. A nonlinearly damped Duffing oscillator is taken as an example to show the application and validity of the method. In the case of primary external resonance, based on the stationary joint probability density of amplitude and phase difference, the stochastic jump of the Duffing oscillator and P-bifurcation as the system parameters change are examined for the first time. The agreement between the analytical results and those from Monte Carlo simulation of original system shows that the proposed procedure works quite well.

Stochastic Response of Nonlinear Viscoelastic Systems with Time-Delayed Feedback Control Force and Bounded Noise Excitation

2021 ◽  
pp. 2150181
Author(s):  
Xudong Gu ◽  
Fusen Jia ◽  
Zichen Deng ◽  
Rongchun Hu
Keyword(s):  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Delayed Feedback Control ◽  
Control Force ◽  
Stochastic Response ◽  
Bounded Noise ◽  
Strongly Nonlinear ◽  
Nonlinear Viscoelastic ◽  
Time Delayed Feedback

In this paper, an approximate analytical procedure is proposed to derive the stochastic response of nonlinear viscoelastic systems with time-delayed feedback control force and bounded noise excitation. The viscoelastic force and the time-delayed control force depend on the past histories of the state variables, which will result in infinite-dimensional problem in theoretical analysis. To resolve these difficulties, the viscoelastic force and the time-delayed control force are approximated by the current state variable based on the quasi-periodic behavior of the systematic response. Then, by using the stochastic averaging method for strongly nonlinear systems subjected to bounded noise excitation, an averaged equation for the equivalent system is derived. The Fokker–Plank–Kolmogorov (FPK) equation of the associated averaged equation is solved to derive the stochastic response of the equivalent system. Finally, two typical nonlinear viscoelastic oscillators are worked out and the results demonstrated the effectiveness of the proposed procedure. By utilizing the quasi-periodic behavior and stochastic averaging method of the strongly nonlinear system, the time-delayed control force and the viscoelastic terms can be simplified with equivalent damping force and equivalent restoring force and the resonant response under bounded noise excitation can be obtained analytically. The numerical results showed the accuracy of the proposed method.

scholarly journals Optimal Bounded Control for Stationary Response of Strongly Nonlinear Oscillators under Combined Harmonic and Wide-Band Noise Excitations

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Yongjun Wu ◽  
Changshui Feng ◽  
Ronghua Huan
Keyword(s):  
Joint Probability ◽  
Wide Band ◽  
Nonlinear Oscillators ◽  
Stochastic Averaging ◽  
Bounded Control ◽  
Stationary Response ◽  
Strongly Nonlinear ◽  
Band Noise ◽  
Wide Band Noise ◽  
Strongly Nonlinear Oscillators

We study the stochastic optimal bounded control for minimizing the stationary response of strongly nonlinear oscillators under combined harmonic and wide-band noise excitations. The stochastic averaging method and the dynamical programming principle are combined to obtain the fully averaged Itô stochastic differential equations which describe the original controlled strongly nonlinear system approximately. The stationary joint probability density of the amplitude and phase difference of the optimally controlled systems is obtained from solving the corresponding reduced Fokker-Planck-Kolmogorov (FPK) equation. An example is given to illustrate the proposed procedure, and the theoretical results are verified by Monte Carlo simulation.

Stability of Rotating Pretwisted, Tapered Beams With Randomly Varying Speeds

1998 ◽  
Vol 120 (3) ◽  
pp. 784-790 ◽  
Author(s):  
T. H. Young ◽  
T. M. Lin
Keyword(s):  
Averaging Method ◽  
Wide Band ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Bending Vibration ◽  
Approximation Solution ◽  
Rotating Speed ◽  
System Parameters ◽  
Sample Stability ◽  
Small Intensity

This paper presents a study of stability of coupled bending-bending vibration of pretwisted, tapered beams rotating at randomly varying speeds. The rotating speed is characterized as a wide-band, stationary random process with a zero mean and small intensity superimposed on a constant speed. This randomly varying speed may result in the existence of parametric random instability of the beam. The stochastic averaging method is used to derive Ito’s equation for an approximation solution, and expressions for stability boundaries of the system are obtained by the second-moment and sample stability criteria, respectively. It is observed that those system parameters which will raise the first natural frequency but not enhance the centrifugal force of the beam tend to stabilize the system.

scholarly journals Stochastic bifurcation analysis in Brusselator system with white noise

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Changzhao Li ◽  
Juan Zhang
Keyword(s):  
White Noise ◽  
Averaging Method ◽  
Planck Equation ◽  
Original System ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Stochastic Bifurcation ◽  
Largest Lyapunov Exponent ◽  
Singular Boundary ◽  
Polar Coordinate

Abstract In this paper, we mainly study the stochastic stability and stochastic bifurcation of Brusselator system with multiplicative white noise. Firstly, by a polar coordinate transformation and a stochastic averaging method, the original system is transformed into an Itô averaging diffusion system. Secondly, we apply the largest Lyapunov exponent and the singular boundary theory to analyze the stochastic local and global stability. Thirdly, by means of the properties of invariant measures, the stochastic dynamical bifurcations of stochastic averaging Itô diffusion equation associated with the original system is considered. And we investigate the phenomenological bifurcation by analyzing the associated Fokker–Planck equation. We will show that, from the view point of random dynamical systems, the noise “destroys” the deterministic stability. Finally, an example is given to illustrate the effectiveness of our analyzing procedure.

An improved energy envelope stochastic averaging method and its application to a nonlinear oscillator

2016 ◽  
Vol 23 (1) ◽  
pp. 119-130 ◽  
Author(s):  
Yaping Zhao
Keyword(s):  
Steady State ◽  
Probability Density Function ◽  
Probability Density ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
System Response ◽  
Mean Square ◽  
Fpk Equation ◽  
The Mean

An improved stochastic averaging method of the energy envelope is proposed, whose application sphere is extensive and whose implementation is convenient. An oscillating system with both nonlinear damping and stiffness is taken into account. Its averaged Fokker-Planck-Kolmogorov (FPK) equation in respect of the transition probability density function of the energy envelope is deduced by virtue of the method mentioned above. Under the initial and boundary conditions, the joint probability density function as to the displacement and velocity of the system is worked out in closed form after solving the averaged FPK equation by right of a technique based on the integral transformation. With the aid of the special functions, the transient solutions of the probabilistic characteristics of the system response are further derived analytically, including the probability density functions and the mean square values. A simple approach to generate the ideal white noise is drastically ameliorated in order to produce the stationary wide-band stochastic external excitation for the Monte Carlo simulating investigation of the nonlinear system. Both the theoretical solution and the numerical solution of the probabilistic properties of the system response are obtained, which are extremely coincident with each other. The numerical simulation and the theoretical computation all show that the time factor has a certain influence on the probability characteristics of the response. For example, the probabilistic distribution of the displacement tends to be scattered and the mean square displacement trends toward its steady-state value as time goes by. Of course the transient process to reach the steady-state value will obviously be shorter if the damping of the system is greater.

The Effects of Correlated Noise on Bifurcation in Birhythmicity Driven by Delay

2018 ◽  
Vol 28 (10) ◽  
pp. 1850127 ◽  
Author(s):  
Lijuan Ning ◽  
Zhidan Ma
Keyword(s):  
Averaging Method ◽  
Harmonic Approximation ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Self Control ◽  
Correlated Noise ◽  
State Variables ◽  
Stationary Probability Density Function ◽  
Fpk Equation ◽  
Control Feedback

We consider bifurcation regulations under the effects of correlated noise and delay self-control feedback excitation in a birhythmic model. Firstly, the term of delay self-control feedback is transferred into state variables without delay by harmonic approximation. Secondly, FPK equation and stationary probability density function (SPDF) for amplitude can be theoretically mapped with stochastic averaging method. Thirdly, the intriguing effects on bifurcation regulations in a birhythmic model induced by delay and correlated noise are observed, which suggest the violent dependence of bifurcation in this model on delay and correlated noise. Particularly, the inner limit cycle (LC) is always standing due to noise. Lastly, the validity of analytical results was confirmed by Monte Carlo simulation for the dynamics.

Discrimination of different temporal envelope structures of diotic and dichotic target signals within diotic wide-band noise

2005 ◽  
pp. 397-403 ◽  
Author(s):  
Steven van de Par ◽  
Armin Kohlrausch ◽  
Jeroen Breebaart ◽  
Martin McKinney
Keyword(s):  
Wide Band ◽  
Temporal Envelope ◽  
Band Noise ◽  
Wide Band Noise

STOCHASTIC HOPF BIFURCATION OF QUASI-INTEGRABLE HAMILTONIAN SYSTEMS WITH FRACTIONAL DERIVATIVE DAMPING

2012 ◽  
Vol 22 (04) ◽  
pp. 1250083 ◽  
Author(s):  
F. HU ◽  
W. Q. ZHU ◽  
L. C. CHEN
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Derivative ◽  
Hamiltonian Systems ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Bifurcation Parameter ◽  
Integrable Hamiltonian Systems ◽  
Fractional Derivative Damping ◽  
Stochastic Hopf Bifurcation

The stochastic Hopf bifurcation of multi-degree-of-freedom (MDOF) quasi-integrable Hamiltonian systems with fractional derivative damping is investigated. First, the averaged Itô stochastic differential equations for n motion integrals are obtained by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Then, an expression for the average bifurcation parameter of the averaged system is obtained and a criterion for determining the stochastic Hopf bifurcation of the system by using the average bifurcation parameter is proposed. An example is given to illustrate the proposed procedure in detail and the numerical results show the effect of fractional derivative order on the stochastic Hopf bifurcation.

On linear filtering under dependent wide-band noise

Stochastics ◽  
1988 ◽  
Vol 23 (4) ◽  
pp. 413-437 ◽  
Author(s):  
A. E. Bashirov
Keyword(s):  
Wide Band ◽  
Linear Filtering ◽  
Band Noise ◽  
Wide Band Noise
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